Improving the robustness of the iterative solver in state-space modelling of guitar distortion circuitry

Abstract

Iterative solvers are required for the discrete-time simulation of nonlinear behaviour in analogue distortion circuits. Unfortunately, these methods are often computationally too expensive for real-time simulation. Two methods are presented which attempt to reduce the expense of iterative solvers. This is achieved by applying information that is derived from the specific form of the nonlin-earity. The approach is first explained through the modelling of an asymmetrical diode clipper, and further exemplified by application to the Dallas Rangemaster Treble Booster guitar pedal, which provides an initial perspective of the performance on systems with multiple nonlinearities.

Full text

Proc. of the 18th Int. Conference on Digital Audio Effects (DAFx-15), Trondheim, Norway, Nov 30 - Dec 3, 2015
IMPROVING THE ROBUSTNESS OF THE ITERATIVE SOLVER IN STATE-SPACE
MODELLING OF GUITAR DISTORTION CIRCUITRY
Ben Holmes and Maarten van Walstijn
Sonic Arts Research Center
School of Electronics, Electrical Engineering and Computer Science,
Queen’s University Belfast
Belfast, Northern Ireland, U.K.
{bholmes02,m.vanwalstijn}@qub.ac.uk
ABSTRACT
Iterative solvers are required for the discrete-time simulation of
nonlinear behaviour in analogue distortion circuits. Unfortunately,
these methods are often computationally too expensive for real-
time simulation. Two methods are presented which attempt to re-
duce the expense of iterative solvers. This is achieved by applying
information that is derived from the specific form of the nonlin-
earity. The approach is first explained through the modelling of
an asymmetrical diode clipper, and further exemplified by applica-
tion to the Dallas Rangemaster Treble Booster guitar pedal, which
provides an initial perspective of the performance on systems with
multiple nonlinearities.
1. INTRODUCTION
In physical modelling of analogue distortion circuitry, the great-
est challenges are typically posed by the modelling of nonlinear
components, such as diodes, triodes, and bipolar junction tran-
sistors (BJTs). In recent literature, this topic has attracted spe-
cific attention in relation to real-time implementation, which ne-
cessitates a sharp trade off between accuracy and efficiency, with
a further possible requirement of parametric control, i.e. allow-
ing on-line updates of the system parameters. Various modelling
paradigms have emerged to meet these demands, including Wave
Digital Filters (WDF) [1, 2], state-space models (including the K-
method and variants thereof) [3, 4, 5, 6], and Port-Hamiltonian
Systems [7]. Each of these appoaches can make use of a precom-
puted lookup table (LUT) that stores the nonlinear behaviour, thus
avoiding the need to solve a multidimensional system of implicit
nonlinear equations on-line (see, e.g. [8]). The downside of the
use of LUTs is that it complicates parametric control, in particular
when dealing with multivariate nonlinearities. One way to address
this is by decomposing the nonlinearity, which significantly re-
duces the computational complexity, although accurate simulation
of complex circuits will require very large table sizes [9]. For uni-
variate cases (i.e. circuits with a single nonlinearity or with mul-
tiple, separable nonlinearities), WDFs are exceptionally suited to
real-time implementation, offering both efficiency and modularity
[10]. However, these properties do not readily extend to modelling
systems with multiple, non-separable nonlinearities, in which case
device-specific simplifying assumptions have to be made to avoid
multivariate root-finding [11, 12].
A more general approach is offered by state-space methods,
but initial formulations were not particularly suited to paramet-
ric control due to the need for computationally expensive matrix
inversions. An elegant solution was offered in [13], proposing
a Nodal DK formulation that employs strategic matrix decom-
position to reduce the inversion costs associated with parameter
updates, without sacrificing the beneficial feature of automated
derivation of the state-space equations. Nevertheless, the approach
still requires numerically solving a system of nonlinear equations,
which is commonly achieved with Newton’s method or variants
thereof. Such iterative methods entail the risk of not converging to
a suitably accurate solution within a limited number of iterations, a
problem that is most prevalent when driving the circuit with signals
of high amplitude and/or frequency, and that is further exacerbated
when increasing the number of non-separable system nonlineari-
ties.
In this paper we present two new adaptations of Newton’s
method which exploit the form of the nonlinear function of the
selected system to help limit the computational cost of finding
the root. A key feature is their amenability to parameter updates
through the use of analytic expressions. The performance of these
methods with the Nodal DK-method is evaluated through compar-
ison with existing root-finding methods in terms of robustness and
computational efficiency.
2. NODAL DK-METHOD
The Nodal DK-method was first developed in [3] to algorithmi-
cally generate state-space models of nonlinear audio circuits. The
method applies Modified Nodal Analysis (MNA) to build a com-
putable system from nodal equations, and uses the trapezoidal rule
to discretise reactive components. The specific method used in this
paper to model circuits is described in [13]. The state space model
is represented by
x[n] = Ax[n−1] + Bu[n] + C f(vn[n]) (1)
y[n] = Dx[n−1] + Eu[n] + F f(vn[n]) (2)
vn[n] = Gx[n−1] + Hu[n] + K f (vn[n]) (3)
where xis the state variable, uis the model input, yis the model
output, and f(vn)represents the terminal currents of the nonlinear
elements relative to the nonlinear voltage vn. Coefficient matrices
A−Hand Kcontrol the linear combinations of each variable
used to update the state and output. The model is updated by first
finding the nonlinear voltage state, which is then used to update
the state variable. To find the nonlinear voltage state, vn, (3) must
be solved numerically. This amounts to finding the root of the
function
g(vn[n]) = p[n] + Kf (vn[n]) −vn[n](4)
where p[n] = Gx[n−1] + Hu[n].
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Proc. of the 18th Int. Conference on Digital Audio Effects (DAFx-15), Trondheim, Norway, Nov 30 - Dec 3, 2015
3. NUMERICAL ROOT FINDING METHODS
Initially, a wide selection of root-finding methods were trialled to
assess which met conditions that suggest real-time capability. The
methods must both: be extendible to multivariate cases, and con-
verge within a specified number of iterations.
In the domain of audio circuit modelling, nonlinear elements
are based upon physical properties. Functions based upon these
properties typically have unique roots, and sufficiently well-
conditioned gradients that many root finding methods utilise. We
therefore define the term non-convergent as a measure of robust-
ness, where for cases on specific computational systems either:
the current value exceeds values representable by normal floating
point arithmetic; or the number of iterations exceeds a limit that
can be completed in an allocated amount of time.
3.1. First Order Methods
Newton’s method uses a linear approximation to the nonlinear func-
tion to successively find better approximations to the root of the
function. Several methods use this technique as a basis, of which
four are discussed. A more comprehensive understanding of these
methods can be obtained from the literature [14].
3.1.1. Newton’s Method
The iterative method employed by Newton’s method is typically
expressed as
vi+1 =vi−J−1(vi)g(vi)(5)
where viand vi+1 are the current and next iterate, g(vi)is func-
tion at the current iterate known as the residual, and J(vi)is the
Jacobian matrix.
To detect when a root has been found, the inequality
|vi+1 −vi|<TOL must be satisfied, which specifies the error
is less than a certain tolerance, represented by TOL. The toler-
ance is selected by the user, and often informed by the required
accuracy of the result, and the system’s numerical precision.
3.1.2. Damped Newton’s Method
By applying damping to Newton’s method, iterations that increase
the residual can be corrected. This is accomplished by reducing
the step size until the residual at the new iterate is less than the
residual at the previous iterate. This is applied to (5) as a scalar
multiplier of the step, so that
vi+1 =vi−2−mJ−1(vi)g(vi)(6)
where the value of mis the smallest integer that satisfies the in-
equality [14]
||gvi−2−mJ−1(vi)g(vi)|| ≤ ||g(vi)||.(7)
The value of mis found by iteratively incrementing the value un-
til the condition is satisfied. Damped Newton’s method has been
shown to be successful for nonlinearities that are more likely to
demonstrate non-convergence, for example BJTs [15, 16].
3.1.3. Chord Method
The most expensive operation in Newton’s method is the calcula-
tion of the inverse Jacobian. To lessen the computational cost of
the method, it is possible to only calculate the Jacobian at the ini-
tial iterate, and use this at each successive iterate. A disadvantage
of this method is that if the Jacobian at the initial iterate causes a
step that overshoots the root, the overshoot is more likely to hap-
pen successively, causing divergence from the root.
3.1.4. Secant Method
The secant method uses a difference method to calculate the Ja-
cobian. In univariate cases, it has been successfully applied in the
simulation of a triode [12]. For multivariate models the method ex-
tends to Broyden’s method. To numerically approximate the Jaco-
bian, Broyden’s method requires an initial Jacobian which it then
updates using a difference method. Upon initial testing, Broyden’s
method was less robust than the Chord method. For this reason it
was not included in the final comparison.
3.2. Quadratic Methods
Halley’s method extends Newton’s method using both Jacobian
and Hessian matrices to form a quadratic approximation to the
nonlinear function. Supporting literature demonstrates that Hal-
ley’s method has faster convergence than Newton’s method [17].
It was found that Halley’s method was less robust than Newton’s
method, and it was for this reason Halley’s method was not in-
cluded in the final comparison.
Brent’s method implements a difference approach to form a
quadratic function [18]. Additional bracketing and conditions are
applied to improve robustness. For univariate cases, this method
proved to be the most robust method, and exhibited good conver-
gence. However, the method has not been extended to multiple
dimensions so was not included within the final comparison.
3.3. A Semi-Analytic Form
The Lambert W function provides analytical solutions for equa-
tions of the form
W(z)eW(z)=z. (8)
This has been applied successfully to diodes with series resistance
both in a general case [19] and using Wave Digital Filters [20].
The Lambert W function is also applicable to state-space mod-
els. It does not extend to multivariate cases and therefore was not
included in the final comparison, but is functional for univariate
cases. This can be shown using a generic circuit featuring a single
diode modelled using the Shockley equation from (12). The non-
linear function from (4) must then be re-arranged into the form of
(8) to find W(z)and z. For the diode case this gives
W(z) = −K(f(vn)) −IS)
NVT
, z =−KIS
NVT
e
p+KIs
NVT.(9)
Where Kis the coefficient from (4) in scalar form. Solving for
f(vn)then yields
f(vn) = −NVT
KW(z)−IS.(10)
An accurate model of anti-parallel diodes can be formed by adapt-
ing (10), taking the absolute value of pand multiplying f(vn)by
sgn(p)[20] to incorporate the polarity. This relies on the assump-
tion in (18), where for this case a≈b. For asymmetrical diodes, a
conditional statement must be applied, incorporating both the po-
larity of f(vn)and the different coefficients in the Shockley equa-
tion.
4. EXPLOITING THE FORM OF THE NONLINEAR
FUNCTION
With respect to the variable vn, the function of g(vn)in (4) can
be decomposed into a constant term, a linear term, and a nonlinear
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Proc. of the 18th Int. Conference on Digital Audio Effects (DAFx-15), Trondheim, Norway, Nov 30 - Dec 3, 2015
term:
g(vn[n]) = p[n] + Kf (vn[n])
| {z }
gn(vn[n])
−vn[n]
|{z}
gl(vn[n])
(11)
where gn(vn[n]), and gl(vn[n]) indicate nonlinear and linear func-
tion components. In this section we propose two new Newton-
based methods that employ system knowledge derived from this
decomposition: the Capped Step and New Iterate methods. These
methods are explained using two case studies, covering both uni-
variate and multivariate nonlinearities. Sound examples and mod-
els can be found at:
http://bholmesqub.github.io/DAFx15/.
4.1. Univariate Case: Asymmetrical Diode Clipper
A univariate nonlinearity is exemplified here by a diode clipper,
which has been covered extensively in the literature [21, 6]. The
circuit uses the exponential nature of the voltage-current relation
of the diode to limit the voltage output. The specific diode clipper
used here can be seen in Figure 1, and features anti-parallel diodes
in a 2:1 ratio.
Vin
R1
C1Vout
Figure 1: Schematic of the modelled asymmetrical diode clipper.
0 1 2 3 4 5
−1
−0.5
0
0.5
Time (ms)
Vout(V)
SPICE
State−space
Figure 2: A2 V,1 kHz sine wave processed by both SPICE and
state-space diode clippers. fs= 176.4 kHz
The Shockley model is used as the component model for the
diodes, representing the current through a diode as
ID=ISe
VD
NVT−1(12)
where ISis the reverse saturation current, VDis the voltage across
the diode, VTis the thermal voltage, and Nis the ideality factor.
Noting in this case vn=VD, the asymmetric combination forms
the nonlinear term
f(vn) = ISe
vn
NVT−1−ISe
−vn
2NVT−1(13)
where the factor of 1/2in the second exponent represents the two
diodes, as each diode carries half of the voltage drop across the ter-
minals. This relies on the assumption that the diodes are identical.
−1.5 −1 −0.5 0 0.5
−4
−3
−2
−1
0
1
2
3
4
vn(V)
Volts
‘‘Linear’’ Region
g(vn)
Kf (vn)
−vn
Vtr
Figure 3: Decomposed regions of the diode clipper nonlinearity
where p[n] = 0,fs= 176.4 kHz.Vtr
+= 0.5052 V, V tr
−=
−1.0731 V.
For this specific model, the following component values were
used: R1= 2200 Ω,C1= 0.01 µF,IS= 2.52 nA,N= 1.752,
VT= 25.8 mV. The diode values are taken from LTspice IV [22],
and refer to a 1N4148 signal diode. The state-space model has
been validated using SPICE, which is illustrated in Figure 2.
4.1.1. Capping the Newton Step
A problematic case for Newton-based methods arises when the
gradient at the initial iterate causes a Newton step that overshoots
the root of the function. The exponential nature of the examined
nonlinear terms prevents this for large values of p[n]. When p[n]
is small, the nonlinear term becomes significantly smaller than the
linear term, which can cause an overshoot if the root is not in close
proximity. In extreme cases, the residual exceeds values repre-
sentable by normal floating point arithmetic. As seen in Section
3.1.2, applying damping to Newton’s method aids this with the
trade-off of sub-iterations.
An alternative approach is to set a maximum step size, for
example with a simple comparative function:
∆vn=(sgn(∆vn)Vlim,|∆vn|> V lim
∆vn,|∆vn| ≤ Vlim (14)
where ∆vnand ∆vnrepresent the capped and unaltered step size,
Vlim is the limit placed upon it, and the signum function adjusts
the polarity. For this to be successful, a limit must be specified
that is large enough to prevent drastically increasing the number
of iterations required. A suitable value is defined by finding the
transitional voltages beyond which the nonlinear term is dominant,
as illustrated in Figure 3 (which also compares the decomposition
of the nonlinear function from (11)). The distance of this voltage
from the origin is applied as the limit, such that Vlim =|Vtr |,
where Vtr is the transitional voltage.
4.1.2. Defining System-Specific Transitional Voltages
To find the transitional voltages of the nonlinear function, the gra-
dient information of the nonlinear and linear terms are compared.
For the univariate case, this amounts to finding the two values of
vnfor which dgl/dvn=dgn/dvn. Applying this using (18) to
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Proc. of the 18th Int. Conference on Digital Audio Effects (DAFx-15), Trondheim, Norway, Nov 30 - Dec 3, 2015
separate terms yields
−1 = KIS
NVT
e
vn
NVT,−1 = KIS
2NVT
e
−vn
2NVT.(15)
Solving these equations for vnfinds the transitional voltages, ex-
pressed as:
Vtr
+=NVTlog −NVT
KIS, V tr
−=−2NVTlog −2NVT
KIS.
(16)
4.1.3. Setting a Strategic Initial Iterate
Typically, the solution from the previous sample is used as an ini-
tial iterate to find the solution at the current sample. Fast conver-
gence then relies on the assumption of small inter-sample differ-
ences, but this breaks down with inputs of high-frequency and/or
amplitude, depending also on the sampling frequency. An alterna-
tive to this is to use an approximation to the nonlinear function,
which will place the initial iterate at a position which prevents
overshoot of the root (as discussed in Section 4.1.1) and is inde-
pendent of the past sample. This forms the basis of the New Iterate
method, which attempts to reduce the dependency of convergence
on the input and sampling frequency.
The proposed approximation to the univariate version of (11)
is formed by removing the linear term, which is accurate when vn
is large. If the nonlinear term f(vn[n]) is an invertible function,
this allows for an analytical solution for vn[n], where the general
univariate form is
vNI
n[n] = f−1−p[n]
K.(17)
To apply this to the asymmetrical diode clipper, the nonlinear func-
tion can be separated into positive and negative terms, using the
assumption ea|vn|−1e−b|vn|−1(18)
where aand bare positive constants. The two separate functions
can then be inverted to solve for the new initial iterate
vNI
n[n] = 


NVTlog 1−p[n]
KIS, p[n]≥0
−2NVTlog 1 + p[n]
KIS, p[n]<0
(19)
where p[n]is used to determine the polarity.
4.2. Multivariate Case: Dallas Rangemaster
To exemplify systems with more than one nonlinearity, the Dal-
las Rangemaster is modelled. The Rangemaster is an early “treble
booster” pedal which increases the amplitude of the guitar signal to
drive the amplifier into further saturation, particularly at higher fre-
quencies. Figure 4 illustrates the complete schematic of the model,
with R4modelling the load of the circuit. The pedal features one
parameter which changes the gain, but for the purpose of compar-
ison it was set to maximum.
The nonlinear behaviour is caused by the PNP BJT, which is
modelled using the Ebers-Moll injection model. The Ebers-Moll
model represents the current through each terminal (Base, Collec-
tor, and Emitter) as a combination of the voltages across its ter-
minals. For a complete model, only two of these equations are
Table 1: Component values of the Rangemaster circuit.
R1470 kΩ R41 MΩ C24.7 pF
R268 kΩ V R110 kΩ C347 µF
R33.9 kΩ C147 µFC410 pF
required as the third can be found using superposition [21]. The
current-voltage relationships can thus be represented by
IB=IS
βFe
VEB
VT−1+IS
βRe
VEB−VEC
VT−1(20)
IC=ISe
VEB
VT−1−IS
βR+ 1
βRe
VEB−VEC
VT−1(21)
where βFand βRare the forward and reverse common-emitter
current gain. The original Rangemaster used a germanium BJT,
but for the model generic parameters were used: IS= 10 fA,
βF= 200,βR= 2 and VTremains the same as for the diode
clipper case. The full nonlinear function is expressed by
g(vn) = p+KIB
IC−VEB
VEC.(22)
The component values are shown in Table 1. The state-space model
was validated with SPICE, which is illustrated in Figure 5. To pro-
duce this result, both simulations were initialised with steady-state
solutions.
Vcc C1
Vin
C2
R2
R1
V R1
R3
C4
Vout
R4
Figure 4: Schematic of the modelled Dallas Rangemaster Treble
Booster.
0 1 2 3 4
−2
0
2
4
6
Time (ms)
Vout(V)
SPICE
State−space
Figure 5: A200 mV,1 kHz sine wave processed by both SPICE
and state-space Rangemasters. Vcc = 9 V,fs= 176.4 kHz.
4.2.1. Setting a Multivariate Initial Iterate
Finding an approximation of a multivariate function follows the
same process as applied to the univariate case. To find the inverted
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Proc. of the 18th Int. Conference on Digital Audio Effects (DAFx-15), Trondheim, Norway, Nov 30 - Dec 3, 2015
form of the Ebers-Moll functions, they must be decomposed. To
accomplish this, the Ebers-Moll functions can be expressed as the
product of a square matrix and a vector:
IB
IC=L
e
VEB
VT−1
e
VEB−VEC
VT−1
,L=IS"1
βF
1
βR
1−βR+1
βR#.
(23)
The simplified nonlinear equation of the nodal DK-method can
then be solved for the vector containing the exponents:
−Q−1p=
e
VEB
VT−1
e
VEB−VEC
VT−1
(24)
where Q=KL. Values for VEB and VEC are then solved for by
separately inspecting the terms in (24), where VEC is found from
the lower term after first determining VEB from the upper term:
VNI
EB =VTlog1−ˆp1, V NI
EC =VNI
EB −VTlog1−ˆp2(25)
where ˆp=Q−1p.
4.2.2. Defining System-Specific Transitional Voltages
Each term of the Ebers-Moll functions depends upon VEB, which
complicates the process of finding independent transitional volt-
ages. By creating a new voltage vector ˆv=VEB VCBTusing
the substitution VCB =VEB −VEC, two voltages are provided of
which to find the transitions. The equation ∂gn/∂ ˆv=∂gl/∂ˆv
is then used to find each transition. Two transitions are found for
VEB,
Vtr
EB =VTlog−VT
Q11 and Vtr
EB =VTlog−VT
Q21 ,(26)
and one transition is found for VCB,
Vtr
CB =VTlogVT
Q22 .(27)
As the solution for Vtr
CB is found using the partial derivative w.r.t.
VCB,VEB is ignored allowing the limit relative to VEC to be de-
fined as Vlim
EC =|Vtr
EC|=| − Vtr
CB|.
4.2.3. Capping the Multivariate Newton Step
To apply capping to a multivariate step, the same function from
(14) can be applied individually to each term. The lower of the
two values from (26) is applied as the limit for VEB. In the case of
the modelled BJT, these values are in close proximity so that the
difference in performance is negligible.
5. COMPUTATIONAL COST
To assess the efficiency of the root-finding methods, they were
compared in terms of the number of operations required to con-
verge. The Lightspeed Matlab toolbox [23] was used to provide
costs of floating point operations (FLOPs). Integer operation costs
were set equal to the floating point equivalent. Branch operations
were given the same cost as logical and relational operators. Con-
trol dependencies were ignored for simplicity as they are difficult
to represent using an operation cost. These choices inform two
specifications about the theoretical hardware used for the simula-
tion: the integer and floating point hardware performs equally, and
there is no instruction level parallelism (i.e. operation pipelining).
The cost of each operation used within the algorithms is stated in
Table 2.
Table 2: Cost of individual operations.
Operation Cost
+,−,×1
logical, relational,
branch
2
abs() 4
sgn() 5
÷8
exp() 40
||x||22M+ 7
Solve using LU M3+1
2M2+29
2M−8
5.1. Method Costs
Using the values and expressions from Table 2, the cost of each
method was determined. Each cost is determined based upon the
number of dimensions it is solving for, M, and the number of it-
erations it performs, i. Additionally, the Damped Newton method
requires sub-iterations, denoted by is. The costs of calls to the
function and Jacobian are represented by CFand CJrespectively.
Clim and Citer represent the initial cost of calculating the transi-
tional voltages and the approximate initial iterate. These values
are found at each time step, assuming each method is applicable to
audio rate parametric control.
The cost of each method is denoted using subscript: CNfor
Newton’s method; CDfor Damped Newton’s method; CCfor the
Chord method; CCS for Newton’s method with the capped step
applied; and CNI for Newton’s method with the new initial iterate.
CN=M3+1
2M2+29
2M+CJ+CF−8
+iM3+1
2M2+35
2M+CJ+CF+ 8(28)
CD=M3+1
2M2+29
2M+CJ+CF−8
+iM3+1
2M2+43
2M+CJ+CF+ 12
+is6M+CF+ 6
(29)
CC=M3+1
2M2+29
2M+CJ+CF−8
+iM3+1
2M2+35
2M+CF+ 8(30)
CCS =CN+ 21M+ 21iM +Clim (31)
CNI =CN+Citer (32)
Table 3 contains the cost of constant values for both the diode clip-
per and the Rangemaster models. Using this information, numeri-
cal values were obtained for the cost of an iteration and the initial
computation for each algorithm. These are displayed in Table 4.
6. RESULTS
Test simulations were designed to compare the performance of
each method against two properties: the amount of oversampling
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Proc. of the 18th Int. Conference on Digital Audio Effects (DAFx-15), Trondheim, Norway, Nov 30 - Dec 3, 2015
Table 3: Cost in operations of constant values for both diode clip-
per and Rangemaster models.
Variable Diode Clipper Rangemaster
Clim 32 124
Citer 37 130
CF105 234
CJ121 359
Table 4: Model-specific cost in operations for the computation
required for one iteration and the initial computation of each
method.
Diode Clipper Rangemaster
Method Init. Iter. Init. Iter.
Newton 234 253 624 646
Damped 234 261 +
117is
624 658 +
252is
Chord 234 132 624 287
New It. 271 253 754 646
Capped 287 274 790 688
applied, and the peak voltage of the input. Oversampling is com-
pared to test how efficient each method is on computational sys-
tems with different processing capabilities.
A 30 period, 1 kHz sine wave was used to drive the models.
The sine wave was modulated by a Hann window so that the ampli-
tude varied across the range of the nonlinearity. For both circuits,
the peak voltage of the input was chosen to match what can be
expected from a real circuit. As a diode clipper is typically situ-
ated after amplification, the highest peak voltage was set at 9 V,
which presumes the system uses a dual-rail ±9 V power supply.
The Rangemaster is designed to be placed at the start of a gui-
tarist’s signal chain, so the input reflects a guitar’s output. For this
reason a representative highest peak voltage was set at 300 mV,
although it is noted guitar output voltages can exceed this. The
power supply voltage for the Rangemaster model, Vcc was set to
9 V.
To ensure a fair comparison, the parameters of the root find-
ing methods were set constant between models and methods. The
tolerance was set to 10−12, and the maximum number of itera-
tions was set to 100. Observed inefficiency of Damped Newton’s
method was corrected by limiting the number of sub-iterations to
3.
Results from the simulations were filtered to emulate the buffer-
ing of a real system. Figure 6 shows an example of the unfiltered
iterations, and the iterations after being processed by a moving av-
erage filter with a window of 2 ms. Table 5 shows results of a set
of 16 simulations. Both maximum iteration and operation counts
are provided, for which a filtered version and unfiltered version
are displayed. Figures 7 and 8 illustrate the performance of the
diode clipper and Rangemaster over a range of amplitudes, with
no oversampling.
The most notable result from these simulations is that both
Chord and Newton’s methods exhibit non-convergent behaviour in
a variety of tests in which the other three methods are convergent.
Of these remaining methods, each has several test cases in which
it is the most efficient.
One exclusive feature is the uniform behaviour of the New
0 5 10 15 20 25
−2
0
2
4
6
8
Vout(V)
Output
Input
0 5 10 15 20 25
0
10
20
30
Iterations
Time (ms)
Unfiltered
Moving Av.
Figure 6: Input/Output and iteration count of a 1 kHz,200 mV
sine wave modulated by a hann window processed by the Range-
master state-space model using Newton’s method, fs= 88.2 kHz.
Unfiltered and moving average filter results shown, and maximum
values marked with .
Iterate method. This is clearly observable from the consistent be-
haviour relative to sampling frequency, with the maximum vari-
ation of 1 iteration (peak) for the case of the Rangemaster with
a peak voltage of 300mV. Figure 7 and 8 confirm this behaviour
relative to input voltage, although with higher variance.
7. CONCLUSIONS
In this paper two novel root-finding methods were presented using
system derived knowledge to improve robustness. The results in-
dicate that for cases of moderate peak voltage and higher sampling
frequency, Newton’s method is sufficiently robust and relatively
efficient. However, for more challenging cases (i.e. cases of high
peak voltage and/or low sampling frequency), Newton’s method
was found to be non-convergent. In principle this can be addressed
by using Damped Newton’s method, although for several tests it
proved to be less efficient than both proposed methods.
The uniform behaviour of the New Iterate method allows the
setting of a fixed number of iterations without risking non-
convergence, thus alleviating control dependencies. This cannot
be achieved by Damped Newton’s method, as a branch instruction
is required to reduce the step size. The Capped Step method can
be configured without control dependencies, but due to its high
variance finding a fixed number of iterations is non-trivial. Con-
trol dependencies were not considered in this paper as they require
focus at a hardware level, but they are known to significantly de-
crease processor performance [24]. This property suggests that
considerable efficiency could be gained using a fixed number of
iterations with a method as opposed to a conventional configura-
tion. To assess the consequences of control dependencies, further
investigation is required.
A key aspect of the proposed iterative methods is that they
rely on the availability of an analytic inverse of either the non-
linear term of the equation to be solved for or its first derivative.
This criterion is generally satisfied since the components in dis-
tortion circuits are normally modelled with monotone analytical
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012345678910
400
600
800
1000
1200
1400
1600
1800 Moving Average
Vpeak(V)
Max Operations
Newton
Chord
Damped
New Iter.
Capped
012345678910
1000
2000
3000
4000
5000
6000 No Averaging
Vpeak(V)
Max Operations
Figure 7: Maximum operations against input gain for the diode
clipper model, no oversampling applied. (Top) The peak averaged
iteration cost (Bottom) The peak iteration cost.
functions. However one possible limitation is that the analytic
inverse function for a specific component model contains signif-
icantly more terms than in the cases presented in this study, which
may then increase the computational costs accordingly. Hence a
further interesting research direction to explore in future research
is to test the methodology on more complex component models.
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Table 5: Results from simulations of both the diode clipper and Rangemaster models, fs= 44.1 kHz. Average notates a moving average
filter has been applied, Peak notates no filtering. Entries marked "-" indicate the method was non-convergent.
1×fs2×fs4×fs8×fs
Average Peak Average Peak Average Peak Average Peak
Its. Ops. Its. Ops. Its. Ops. Its. Ops Its. Ops. Its. Ops. Its. Ops. Its. Ops.
Diode Clipper, Vpeak = 1 V
Newton 3.2 1038 5 1499 2.8 953 4 1246 2.6 896 3 993 2.3 810 3 993
Damped 3.2 1064 5 1539 2.8 976 4 1278 2.6 917 3 1017 2.3 828 3 1017
Chord 9.1 1434 48 6570 5.7 981 15 2214 4.2 787 8 1290 3.4 679 6 1026
New It. 5.5 1670 6 1789 5.5 1656 6 1789 5.4 1644 6 1789 5.4 1631 6 1789
Capped 3.2 1158 5 1657 2.8 1066 4 1383 2.6 1004 3 1109 2.3 911 3 1109
Diode Clipper, Vpeak = 4.5 V
Newton 4.0 1240 13 3523 3.3 1080 8 2258 3.0 982 5 1499 2.7 927 4 1246
Damped 3.8 1233 7 2295 3.3 1100 6 1917 3.0 1005 5 1539 2.7 949 4 1278
Chord - - - - - - - - 6.8 1132 99 13302 4.7 854 17 2478
New It. 5.5 1667 6 1789 5.6 1685 6 1789 5.7 1718 6 1789 5.8 1742 6 1789
Capped 4.0 1371 12 3575 3.3 1203 8 2479 3.0 1097 5 1657 2.7 1038 4 1383
Rangemaster, Vpeak = 100 mV
Newton 2.9 2518 3 2562 2.8 2442 3 2562 2.6 2308 3 2562 2.3 2091 3 2562
Damped 5.2 5085 11 12902 3.9 3619 7 6994 3.1 2769 5 4670 2.3 2185 4 3508
Chord 5.7 2259 8 2920 4.5 1911 6 2346 3.8 1703 5 2059 3.4 1587 4 1772
New It. 8.4 6176 9 6568 8.4 6177 9 6568 8.4 6156 9 6568 8.0 5922 9 6568
Capped 2.9 2808 3 2854 2.8 2726 3 2854 2.6 2583 3 2854 2.3 2352 3 2854
Rangemaster, Vpeak = 300 mV
Newton - - - - - - - - 2.8 2427 41 27110 2.5 2241 19 12898
Damped 6.4 7013 19 23962 5.0 5122 20 25124 3.9 3852 23 28610 3.0 2867 22 27448
Chord - - - - - - - - - - - - - - - -
New It. 8.4 6169 12 8506 8.4 6177 13 9152 8.4 6151 13 9152 8.0 5922 13 9152
Capped 3.8 3434 26 18678 3.2 3006 21 15238 2.7 2667 21 15238 2.5 2510 13 9734
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